The heads of the security services should not be directly accountable to parliament, the former GCHQ chief has said. Sir David Omand said they should make more public appearances to make it easier to see "the kind of people they are", but that accountability was a different matter and would "build up the agency heads into something they are not".

Omand was speaking at a debate on spying held in parliament on Tuesday evening, and was joined by Wikipedia co-founder Jimmy Wales, the Guardian's editor-in-chief Alan Rusbridger and Labour MP George Howarth, a member of the intelligence and security committee, which oversees the work of the UK security services. Shadow home affairs minister Diana Johnson and her Labour colleague Katy Clark also took part in the debate on Tuesday evening.

Omand said: "I think it's helpful for the intelligence agency heads to be seen in public. It gives the public a chance to actually look them over and see what kind of people they are.

"But accountability is something else again. It's ministers who are accountable to parliament; it is not the agency heads who are accountable to parliament. [There are] constitutional arrangements and you're building up the agency heads into something they are not. They report to ministers and ministers report to parliament."

Wales said the public should be told whether or not the security services were storing their emails and listening to their phone calls. He acknowledged that doing so may provide some level of help to Britain's enemies "but that's tough – too bad", he said.

He said he speaks to leaders in what he said were authoritarian countries, such as China, and that he implores them to be more open. But, he said, the recent revelations about snooping allowed them to point to everything the UK and US were doing and say "what are you talking about?"

But Omand argued that intelligence agencies should be allowed access to technology companies via the "back door". He told the audience: "I shouldn't really talk about any of this, but just remember the principle: if you put a back door into something you secure it with encryption that is stronger than the system that you are putting the back door into. You are not, therefore, leaving it open."

Clark said the state did not have a very good track record on civil liberties issues, irrespective of which government was in power.

She said the mass surveillance techniques being used were employed "not in our name and not with our consent" and that history showed that targeted surveillance was more effective than mass surveillance. Rusbridger said the debate about the security services' methods was broader than the "binary terms: liberty v security". He listed ten issues he believed it was in the public interest to debate, including the consent of citizens to have their data collected; the legality of the security services' actions; and the involvement of the private sector.

He said that the information leaked by Edward Snowden provoked vital conversations about the risk of the intelligence agencies' information gathering to the digital economy and the deliberate weakening of the security of the internet by UK and US security services.

Rusbridger added that the leaks had led to debates over whether the databases being kept were necessary and effective and whether politicians had the necessary knowledge to exercise proper oversight.